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Popoviciu type characterization of positivity of sums and integrals for convex functions of higher order

Some very general identities of Abel and Popoviciu type for sums $\sum p_i f(x_i)$, $\sum \sum p_{; ; ; ; ; ij}; ; ; ; ; f(x_i, y_j)$ and integral $\int \int P(x, y) f(x, y) dx \, dy$ are deduced. Using obtained identities, positivity of these expressions are characterized for convex functions of higher order. An application in terms of exponential convexity is given.

convex function; $\nabla$-convex function; exponential-convexity; mean

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